Mathematical Sciences: Vortices in Algebraic and Symplectic Geometry

该奖项支持在代数和辛几何中的涡旋领域的研究。它是基于研究者在涡旋和稳定全纯对方面的早期工作。旋涡是Yang-Mills-Higgs能量的极小值，而全纯对是具有指定全纯截面的全纯向量丛。这项研究涉及到涡旋和稳定对研究的新方向，并将对的概念推广到由两个全纯丛和一个全纯丛同态组成的三元组。这将解决通过黎曼曲面与射影直线的笛卡尔乘积上的等变稳定丛的降维来获得对象的方法。这一奖项是在几何学的一般领域，特别是在代数和辛几何领域。这项研究将有助于我们对四维流形的理解，也是当前“规范理论”研究的核心。在外行术语中，“流形”扮演着表面的角色。

This award supports research in the area of vortices in algebraic and symplectic geometry. It is based on earlier work by the investigator on vortices and stable holomorphic pairs. Vortices are minima of Yang-Mills-Higgs energy, and holomorphic pairs are holomorphic vector bundles with prescribed holomorphic sections. This research concerns new directions in the study of vortices and stable pairs and involves a generalization of the notion of a pair to that of a triple consisting of two holomorphic bundles together with a holomorphic bundle homomorphism. This will address the way in which objects can be obtained by a dimensional reduction of an equivariant stable bundle on the Cartesian product of the Riemann surface with the projective line. This award is in the general area of geometry and, in particular, in algebraic and symplectic geometry. This research will aid in our understanding of 4-dimensional manifolds and are central to the current research in 'gauge theory'. In lay terms, a 'manifold' plays the role of a surface.